CRS-STACK-BASED SEISMIC IMAGING AS A CASE STUDY FOR …

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CRS-STACK-BASED SEISMIC IMAGING AS A CASE STUDYFOR BASIN REEVALUATION

Z. Heilmann, L. Leite, and A. Gomes

email: [email protected]: CRS-stack-based imaging, kinematic wavefield attributes, surface topography, residual static

corrections, tomographic inversion, depth migration

ABSTRACT

In this case study we are giving attention to the seismic processing and interpretation of a land data setfrom the Takutu basin, Brazil. This aims at establishing a new seismic imaging workflow well suitedto the specific problems of land data processing, namely, sparse data, complex geological structures,and complicated near surface conditions. The presented extension of CRS-stack-based time-to-depthimaging supports arbitrary top-surface topography. We applied the following processing steps: CRSstack, residual static correction, determination of a macrovelocity model via tomographic inversion,and Kirchhoff pre- and poststack depth migration. Our CRS stack implementation combines twodifferent approaches of topography handling to a cascaded processing strategy that demands verylittle additional effort. After initial values of the CRS attributes are determined for a smoothly curvedmeasurement surface, the final CRS stack and also the CRS-stack-based residual static correctionare applied to the original prestack data, considering the true source and receiver elevations. Thisis very important to achieve physical meaningful CRS attributes. Conventional elevation statics thatassume every ray to emerge vertically deteriorate the CRS attributes, similar as in case of the stackingvelocity. After the CRS stack is completed, we utilize the extracted emergence angles to relate thestacked zero-offset section, the CRS attribute sections and the quality control sections to a chosenplanar reference level. This redatuming procedure removes the influence of the rough measurementsurface from the CRS stack results and provides standardized input for interpretation, tomographicinversion, and poststack depth migration. As a final step, we applied a prestack depth migration fromtopography using the prestack data after residual static correction.

INTRODUCTION

In the current situation of rapidly growing demand in oil and gas, on-shore exploration, even under dif-ficult conditions, becomes again more and more important. Unfortunately, rough top-surface topographyand a strongly varying weathering layer often result in poor data quality, which makes conventional dataprocessing very difficult to apply. Under such circumstances, where simple model assumptions may fail,it is of particular importance to extract as much information as possible directly from the measured data.Fortunately, the ongoing increase in available computing power makes so-called data-driven approaches(see, e. g., Hubral, 1999) feasible which, thus, have increasingly gained in relevance during the last years.The Common-Reflection-Surface (CRS) stack (see, e. g., Mann, 2002) is one of these promising methods.Besides an improved zero-offset simulation, its decisive advantage over conventional methods is that forevery zero-offset sample several so-called kinematic wavefield attributes are obtained as a by-product ofthe data-driven stacking process. As will be shown, they can be applied both, to improve the stack itself andto support subsequent processing steps. Using theseCRS attributes, an advanced data-processing workflowcan be established leading from time to depth domain, covering a broad range of seismic reflection imagingissues in a consistent manner. The major steps of this workflow are displayed in Figure 1.

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144 Annual WIT report 2005

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Figure 1: Major steps of seismic-reflection data processing in time and depth domain. Imaging proceduresthat can be incorporated in the CRS-stack-based imaging workflow are highlighted in yellow.

TAKUTU DATASET

The Takutu dataset used in the case study was acquired by PETROBRAS(Takutu Basin, Rondônia, Brazil)for petroleum exploration. The data is free for use on university research and it was obtained from ANP(Agência Nacional do Petróleo). The goal here is to support academic projects that can deal with basinreevaluation based on seismic reprocessing. The software used is non-commercial, developed in the spiritof continuous cooperation between the Geophysics Department of the Federal University of Pará (Brazil)and the Wave Inversion Technology (WIT) Consortium of the University of Karlsruhe, Germany. Thisdata set is offered in the form of non-processed field records, therefore a complete preprocessing stage wasnecessary.Following the description by Eiras and Kinoshita (1990), the Takutu basin is classified as a Mesozoic in-tracontinental rift, oriented NE-SW, with approximately 300 km length and 40 km width. It was developedin the central part of the Guyana shield, and it is located at the border between Brazil and Guyana. The riftis filled with sediments ranging from the Jurassic to the Quaternary, and composed of two asymmetricalhalf-grabens: The SW part dips southeasterly and the NE part dips northwesterly.The structural scenario of the Takutu basin features horsts, grabens, anticlines, synclines, flower structures,and dip inversions (rollovers). Transcurrent faulting is considered to have reactivated local features thatwere developed in the rift stage (Figure 2).The stratigraphic scenario of the Takutu basin is divided into four depositional sequences that reflect thegeological evolution of the area (Figures 3, 4 and 5). The first basal sequence is represented by the volcanicApoteri formation and by the shaly Manari formation, both related to the pre-rift phase. The second se-quence is represented by the evaporitic Pirara formation, and relates to the stage of maximum stretching inthe rift phase. The third sequence is represented by the sands and conglomerates of the Takutu and Tucanoformations, and is interpreted to correspond to the continuous decrease in stretching. The fourth sequenceis represented by the lateritic and alluvium of the Boa Vista and North Savannas formations.Continuing with Eiras and Kinoshita (1990), the conclusions for the model of the Takutu basin were for-mally based on the interpretation of the conventionally processed seismic data, seismic reprocessing, seis-mic stratigraphy, surface geology, drilled wells, geochronology and geochemistry. Several structural styleswere considered for the basin in focus, and the most attractive were deltaic fan-shaped, compressionalinversions, internal horst highs, and dip reversals. Our intention at this moment is not yet to trace new evi-dences for the structural scenario for the Takutu basin; this may follow with the course of the studies with

Annual WIT report 2005 145

Figure 2: Structural map of Takutu graben redrew from Eiras and Kinoshita (1990) showing the directionsof the seismic lines.

more systematically processed data completing at least a full block for a proper geological interpretation.Among the processed lines, the selected line for this case study was the one numbered 204-239. It hasthe following survey information: date of acquisition 1986; direction NW-SE; length of 31.5 km; 631 shotpoints; 4 ms of time sampling interval; 50 m spacing of shot points and stations; charges of 0.9 kg at 2 mdepth distributed as L-3x2/25 m. The array distribution from left to right starts with a part right-unilateral0-48; the second part is split-spread symmetrical 48-48; the third part is split-spread asymmetrical 76-20;and the fourth part is a left-unilateral 76-0. Unfortunately, the data quality from 20 km to 31.5 km distanceis very bad so that hardly any continuous reflection event could be imaged. Consequently, we had to limitthe target area for this case study to the range between 0-20 km.The preprocessing steps were performed with the CWP/SU package of the Colorado School of Mines (Co-hen and Stockwell, 2000), whose data format is also used in the CRS code. Similar procedures should becarried out for all the lines of the two seismic blocks (numbered 50 and 204) of the Takutu graben. Thetasks consisted of 3 main parts: (1) Geometry setting; (2) Muting of bad traces (3) F and F-K filtering (4)Trace balancing.As a first observation about the original Takutu seismic land-data, the line 204-239 has many noisy sec-tions. For this reason, several shot and receiver gathers were initially completely muted. Afterward, as aresult of visual analysis of all shot gathers, again several single traces had to be zeroed due to the high noiselevel like spikes and sensor wandering. As a second observation, several band-pass filters with polygonalform were experimented, and the decision was for adopting the case with corners 8-10-35-45 Hz. The F-Kvelocity dependent filter was used to further emphasize the cutting of surface waves and critically refractedwaves. The decision for adopting the filter parameters was based on the trace gathers analysis through thespectrum and preliminary stack results, reinforcing the importance of the preprocessing stage on the CRSstack results. Also, it should be mentioned that no deconvolution was carried out on this data.


Figure 3: Seismic section A from Eiras and Kinoshita (1990) with aspects of geological interpretation forTakutu graben.

Figure 4: Seismic section D from Eiras and Kinoshita (1990) with aspects of geological interpretation forTakutu graben.


Figure 5: Seismic section E from Eiras and Kinoshita (1990) with aspects of geological interpretation forTakutu graben.

CRS STACK FOR TOPOGRAPHY

The simulation of stacked ZO sections is routinely applied to enhance the signal-to-noise ratio and reducethe amount of seismic data for further processing. A conventional approach to achieve this goal is theapplication of normal moveout (NMO) and dip moveout (DMO) corrections to the multicoverage datasetfollowed by a subsequent stack along the offset axis, usually denoted as NMO/DMO/stack (e. g., Yilmaz,2001). The CRS stack (e. g., Müller, 1999; Jäger et al., 2001; Mann, 2002) is a powerful alternative to thisconventional approach that can be seen as a generalized multi-dimensional high-density stacking-velocityanalysis tool. It has been reported by several case studies, e.g. Gierse et al. (2003), that the CRS stackproduces reliable stack sections with high resolution and signal-to-noise ratio. As mentioned before a setof physically interpretable stacking parameters is determined as a result of the data-driven stacking process.These kinematic wavefield attributes obtained by the CRS stack are important because they can be appliedto solve a number of dynamic and kinematic problems (see Figure 1). So far, CRS-stack based seismicimaging was limited to data acquired on a planar measurement surface or at least to data for which a planarmeasurement surface had been simulated by elevation statics. However, conventional elevation statics mayintroduce a certain error to the stack and—even worse—to the CRS attribute sections, as a vertical emer-gence of all rays has to be assumed. In case of rough top-surface topography this error might significantlydeteriorate the results of the CRS stack and of all processing steps based on it. Therefore, we extended theexisting implementations of CRS stack and CRS-stack-based residual static correction to consider the truesource and receiver elevations (Heilmann, 2003; von Steht, 2004; Koglin, 2005).

In recent years, two different CRS stacking operators that consider the top-surface topography havebeen developed at Karlsruhe University [see Appendix A]. Chira et al. (2001) assumed a smoothly curvedmeasurement surface for which the elevation of all source and receiver points contributing to a single stack-ing process can be approximated by a parabola. This approach is attractive from the computational point ofview as it allows to adopt most parts of the conventional CRS stack implementation. In particular, the prag-matic CRS-attribute search strategy using three one-parameter searches to determine the optimal stackingoperator can be maintained. However, small elevation statics are still required in order to transfer the realdata to the chosen smoothly curved measurement surface. Zhang (2003) presented a very general CRSstacking operator that directly considers the true elevation of every source and receiver. This approachdemands far more computational effort, as at least two of the three CRS attributes have to be searchedfor simultaneously due to the higher complexity of the stacking operator. On the other hand, no elevation


=> Prestack data related tosmoothed measurement surface

Datum Static Correction

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stack and coherence sections

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=> Elevation, dip, and curvatureof smoothed measurement surface

Topography Analysis

Figure 6: Cascaded processing scheme for topography handling in CRS stack and CRS stack based resid-ual static correction

statics are required and the elevations of the emergence points of the simulated zero-offset rays can bechosen—within certain limits—arbitrarily. A similar approach based on the methology ofMultifocusingwas presented in Gurevich et al. (2001).Following the idea of a step-by-step refinement, we chose an implementation that combines both methodsof topography handling mentioned above to a cascaded processing strategy. Doing this, most of the spe-cific disadvantages of the single approaches can be compensated without loosing their individual benefits.In a final step, we relate the CRS stack results to a planar reference datum close to the actual measure-ment surface. By this redatuming procedure a seamless transition to the tomographic inversion and othersucceeding processing steps is provided. A flowchart of this pragmatic strategy is depicted in Figure 6.

Initial CRS stack

As shown by Heilmann (2003), the CRS traveltime operator for smoothly curved topography is well suitedto determine initial values of the CRS attributes. Therefore, the global search is splitted into three one-parameter searches. Each parameter is determined in a specific 2D subset of the 3D data volume. In orderto apply the CRS traveltime operator for a smoothly curved measurement surface to our data set we have(1) to find an appropriate smooth reference surface, (2) to determine the local dip and curvature in everyCMP location, and (3) to apply datum static corrections that relate the prestack data to this surface.For this purpose, it is necessary to consider that, in general, the larger the scale of the smoothing the largerthe elevation static correction which has to be applied. Nevertheless, the surface has to be smooth enoughsuch that for every single stacking process the elevations of all contributing sources and receivers can bewell approximated by a parabola. To control the degree of smoothing, a smoothing window was introduced.As a rule of thumb, its size should be in the range of the maximum offset aperture used for the wavefieldattribute search and the stacking procedure. A comparison between the original measurement surface andits smoothed counterpart can be found in Figure 7.


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Figure 7: Comparison between original and smoothed measurement surface. The horizontal redatuminglevel atz = 90 m is also displayed.

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Figure 8: ZO section achieved by initial CRS stack considering a smooth curved measurement surface

The datum static correction for every source and receiver location is calculated from the near-surfacevelocity and the differences in elevation of the true source and receiver points and their vertical projectionsonto the smoothed surface. In this specific case we can assume a laterally constant near-surface veloc-ity, i. e. the replacement velocity used in the preprocessing. For every zero-offset sample the kinematicwavefield attributesKNIP, KN, andβ0 are determined by means of three one-parameter searches. Thus,an individual stacking operator (13) is defined for every sample of the zero-offset section. The summa-tion process is performed within a user-defined aperture or within an aperture that accounts for the sizeof the projected first Fresnel zone for offset zero. The latter can be estimated from the CRS attributesby comparing the approximated traveltime of the actual reflection event with the approximated traveltimeof its associated diffraction event (characterized by the propertyKN → KNIP). The locations where thezero-offset traveltimes of these events differ by the temporal wavelet length define the extension of theconsidered projected first Fresnel zone and, thus, the optimum aperture to apply the attribute search andthe stack. The initial CRS stack result achieved in this step is depicted in Figure 8.

Event-consistent smoothing

At this stage, event-consistent smoothing of the initial attributes can be helpful to remove fluctuations andoutliers. The latter are mainly caused by the limitations of the utilized three-times-one parameter search


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Figure 9: ZO section after event-consistent smoothing and three-parameter optimization.

strategy. The smoothing algorithm itself is based on the combined application of mean and median filteringin volumes aligned with reflection events. The information about slopes of events in the time domain isreadily available from the CRS attributes themselves and allows to avoid the mixing of intersecting events.By this means, the smoothing algorithm can be designed in an event-consistent manner. This process doesnot imply any loss of information about the parameterized reflection events (see, e. g., Mann and Duveneck,2004). Thus, even in case of conflicting dip situations, it often provides significantly improved input forthe subsequent optimization. In case of complex near-surface conditions which often lead to a stronglyvariable data quality along the line, this intermediate processing step was especially successful.

Three-parameter optimization considering the true topography

An important feature of the CRS traveltime operator for arbitrary topography (2) is that the elevation ofthe emergence pointX0 of each zero-offset ray to be simulated can be chosen arbitrarily. This propertyprovides the link to the initial results determined under the assumption of a smoothly curved measurementsurface. Since the latter are related to a fictitious smoothly curved measurement surface we choose thissurface also as reference level for the optimization. By this means, the zero-offset rays to be simulatedare identical in both cases. Therefore, it can be expected that the CRS attributes obtained in the previousstep are close to their optimum values and, thus, well suited as initial values for a local three-parameteroptimization using equation (2). The objective function to be maximized is, as in case of the one-parametersearches, the semblance (Neidell and Taner, 1971), a measure of the coherence of the prestack data alongthe stacking operator. Particularly in noisy areas, a significantly increased continuity of the reflection eventscan be expected.To confine the spatial extent of the stacking operator, the projected first Fresnel zone is taken into account.Both, the local optimization and the stacking itself are applied to the original, i. e., uncorrected, data usingthe CRS traveltime operator for arbitrary topography. Consequently, any inaccuracies of the initial stackand especially of the initial attributes caused by the elevation statics should be compensated. Nevertheless,stack and attribute sections are still related to the floating datum corresponding to the smooth referencesurface. The resulting stack section after event-consistent smoothing of the CRS attributes and three-parameter optimization is depicted in Figure 9.

Redatuming

Floating datum sections are no appropriate input for interpretation or further processing. Therefore, weimplemented a redatuming procedure that relates the CRS stack results to a fictitious horizontal measure-


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Figure 10: ZO section after two iterations of residual static correction and redatuming to horizontal refer-ence level atz = 90m

ment surface close to the smoothly curved reference level (see Figure 7). Due to the fact that the emergenceangle of every simulated zero-offset ray is known, it is easy to continue them to a constant reference level—especially, if the replacement velocityvr between the smoothly curved and the planar surface is chosen toequal the near-surface velocityv0 [see Appendix B]. In this case no refraction has to be considered whencrossing the smoothly curved reference level.

RESIDUAL STATIC CORRECTION

The CRS-stack-based residual static correction methodology (Koglin, 2005) is an iterative process close tothe super-trace cross-correlation method by Ronen and Claerbout (1985). However, the cross correlationsare performed within CRS supergathers of moveout corrected prestack traces instead of being confinedto single CMP, common-shot, or common-receiver gathers. The moveout correction makes use of thepreviously obtained CRS attributes and considers the true source and receiver elevations. Thus, elevationstatic correction can be omitted that may introduce, in certain cases, non surface-consistent errors of thesame scale as the searched-for residual statics.Due to the spatial extent of the CRS operator, the CRS supergathers contain many neighboring CMPgathers but with different moveout corrections depending on the current zero-offset location. Thus, thecross correlations of the stacked trace (here used as pilot trace) and the moveout corrected prestack tracesare summed up for each shot and receiver location. This summation is performed for all CRS supergatherscontained in the specified target zone. After all cross correlations are summed up, the searched-for residualtime shifts are expected to be associated with maxima in the cross correlation stacks. Finally, the estimatedtime shifts are used to correct the prestack traces and the next iteration of residual static correction can bestarted. Now, the entire CRS stack process is repeated using the corrected prestack data set. The optimizedCRS stack section after two iterations of residual static correction and redatuming is depicted in Figure 10.It can be observed that besides an strongly improved resolution and event continuity also the small scaleundulations of the reflection events are mostly removed.

TOMOGRAPHIC INVERSION

In order to obtain a depth image from the time-domain pre- and/or poststack data, a macrovelocity modelneeds to be estimated, which is one of the crucial steps in seismic data processing. Fortunately, sucha model can be obtained directly from the CRS stack results: the attributesRNIP andβ0 related to thehypothetical normal-incidence-point (NIP) wave at a given ZO location describe the approximate multi-


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Figure 11: Macro-velocity model [km/s] obtained by CRS attribute based tomographic inversion.

offset reflection response of a common-reflection-point (CRP) in the subsurface. Therefore, the NIP wavefocuses at zero traveltime at the NIP if propagated into the subsurface in a correct model. This principlecan be utilized in an inversion that uses the above mentioned attributes picked in the CRS-stacked sectionto obtain a laterally inhomogeneous velocity model. The CRS-stack-based velocity model determinationapproach is realized as a tomographic inversion (Duveneck, 2004), in which the misfit between picked andforward-modeled attributes is iteratively minimized in the least-squares sense. Fortunately, the tomographicinversion has not to consider the real topography, as the redatuming procedure of the CRS stack providesstack and attribute sections related to a planar reference level.In this case study, about 1000 ZO samples together with the associated attribute values were picked foreach profile to achieve an appropriate resolution and reliability. Automatic picking was performed usinga module based on the coherence associated with the ZO samples. The picked data was checked usingseveral criteria, in order to discriminate outliers and attributes related to multiples, before the tomographicinversion process was applied. The obtained macrovelocity model is defined by 800 B-spline knots. It isdisplayed in Figure 11.

DEPTH MIGRATION

A Kirchhoff type prestack depth migration (PreSDM) procedure for migration from topography (Hertweckand Jäger, 2002; Jäger et al., 2003) was applied. Therefore we used the prestack data after residual staticcorrection together with the macrovelocity model obtained by the tomographic inversion (Figure 11). Thenecessary kinematic Green’s function tables were calculated by means of an eikonal solver. The resultingdepth-migrated prestack data was firstly muted to avoid excessive pulse stretch for shallow reflectors andthen stacked in offset direction in order to obtain the depth-migrated image displayed in Figure 12. Somecommon-image gathers (CIGs) are displayed in Figure 14, where the muting can directly be seen. As mostof the events in the CIGs are flat, we can state that the estimated macrovelocity model is kinematicallyconsistent with the data. Note that no velocity model refinement was applied after the PreSDM.

As a complementary or alternative step of the CRS-stack based imaging workflow, poststack depthmigration (PostSDM) was performed. Input for the PostSDM were the final CRS-stacked section afterredatuming (Figure 10) and the macrovelocity model derived from the CRS attributes (Figure 11). Theresult is depicted in Figure 13.


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Figure 12: Prestack depth migration result.

The PostSDM result as well as the PreSDM result shows many structural details; in particular, faults,vertical offsets of reflectors, deflection of reflectors, changes of reflector characteristics across faults, andfracturing are directly observable in the sections. Although the PreSDM seems to provide a higher res-olution and more details, there are also regions, especially in the deeper part, where some structures arebetter resolved in the PostSDM. Consequently, the PostSDM result provides complementary informationand both migrated sections were used for a structural interpretation.

CONCLUSIONS

We presented results of CRS-stack-based time-to-depth imaging applied to a challenging land-data set.The work was conducted within the framework of an ongoing basin-reevaluation project. The high qualityof the obtained results proves the great potential of CRS-stack-based imaging in complex land-data pro-cessing. Due to the fact that the spatial CRS stacking operator fits the searched-for reflection events inthe prestack data much better than, e.g., the NMO/DMO operator an excellent signal-to-noise ratio canbe achieved. Furthermore, the large fold of the CRS operator provides a significantly increased statisticalbasis for residual static correction making this process more precise and stable. For both, stack and residualstatic correction, no datum static correction of the prestack data is necessary as the topography is directlyconsidered. A cascaded strategy for stacking and attribute extraction was presented combining two differ-ent approaches of topography handling to a very efficient implementation. The obtained CRS attributescan be used to redatum the CRS stack results so that a standardized input for further processing steps isprovided. Finally, a smooth macrovelocity model well suited for subsequent depth migration is determinedfrom the CRS attributes by means of tomographic inversion. In this way a consistent and fully data-drivenimaging workflow can be established leading from time to depth domain.

The geological interpretation should be carried out mainly on the basis of Figures 10, 11, 12, and 13.Doing this, thinning , a long anticline and different faults can be mapped where plays of horsts, grabens androllovers can be indicated. The quality of the Takutu seismic data was the main limitation for imaging theselected line. Nevertheless, the obtained results provide good ground for giving continuity to the Takutuseismic data reprocessing and reinterpretation project. Our intention is to process in the course of thisongoing project also neighboring lines in order to achieve a better geological interpretation of this partof the Takutu graben. We hope that this example serves to reinforce our perspectives and intentions onresearch collaboration between different universities, and between university and industry. Only in thisway a continuous development of seismic exploration technology and of the "human resources" needed in


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Figure 13: Poststack depth migration.

future research and production can be maintained.

ACKNOWLEDGMENTS

The authors would like to thank the Brasilian institution UFPA (Universidade Federal do Pará), FINEP(Financiadora de Estudos e Projetos), ANP (Agência Nacional do Petróleo), PETROBRASand the projectRede Risco Exploratório (Rede 01/03, FINEP 22.01.0763.00) and the sponsors of the WIT (Wave InversionTechnology) Consortium.

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Zhang, Y. (2003).Common-Reflection-Surface Stack and the Handling of Top Surface Topography. LogosVerlag, Berlin.

APPENDIX A

Based on the ray-theoretical foundations presented inCervený (2001), Zhang (2003) derived a very generalCRS stacking operator for arbitrary top-surface topography. This stacking operator describes the second-order traveltime moveout along any ray in the paraxial vicinity of a chosencentral rayby means of phys-ically interpretable properties. These properties, called kinematic wavefield attributes, are directly relatedto the central ray. For comparison, in case of the well known common-midpoint (CMP) stack we haveonly one kinematic wavefield attribute, i. e., the stacking velocity. For the specific geometry of CMP ex-periments on a planar measurement surface this single parameter is sufficient to describe the traveltimemoveout with offset up to the second order. In contrast to this, the CRS stack operator also takes neighbor-ing CMP gathers into account, i. e., it describes the traveltime moveout with offset and midpoint dislocation.

In case of the zero-offset CRS stack, we choose the central ray to have the coincident source and receiverlocationX0 and normal incidence on the reflecting interface. If we denote the traveltime along a paraxial

http://www-gpi.physik.uni-karlsruhe.de/pub/wit/Downloads/diplthesis-mvsteht.pdf


ray by t and the traveltime along the central ray byt0, the so-called parabolic CRS stacking operator forarbitrary topography reads

tpar(∆~m,~h) = t0 −2v0

(∆mx sinβ0 + ∆mz cosβ0)

+KN

v0(∆mx cosβ0 −∆mz sinβ0)

2

+KNIP

v0(hx cosβ0 − hz sinβ0)

2, (1)

where(∆mx,∆mz) and(hx, hz) are the components of midpoint displacement∆~m = ~m − ~OX0 andhalf-offset~h of the considered paraxial ray. The searched-for CRS attributes are:

1. β0, the emergence angle of the central ray measured with respect to thez-axis,

2. KNIP, the curvature of thenormal-incidence-point wavefront, and

3. KN, the curvature of thenormal wavefront.

The mentioned wavefronts are related to hypothetical eigenwaves firstly introduced by Hubral (1983). Theparameterv0 defines the near-surface velocity and is assumed to be known. By taking the square on bothsides of equation (1) and keeping only terms up to second order, we obtain the hyperbolic representation,

t2hyp(∆~m,~h) =(t0 −

2v0

(∆mx sinβ0 + ∆mz cosβ0))2

+2 t0 KN

v0(∆mx cosβ0 −∆mz sinβ0)

2 (2)

+2 t0 KNIP

v0(hx cosβ0 − hz sinβ0)

2.

After systematic investigations, Ursin (1982) suggested that in most cases a hyperbolic traveltime formulais a better approximation to the real traveltime response than a parabolic one. This was later verified by thework of Höcht (1998), Müller (1999), Jäger (1999), and Bergler (2001). Therefore, we use the hyperbolictraveltime expansion for our current implementation.

Even though equations (1) and (2) are quite compact and provide an accurate and very natural descrip-tion of the topography, the computational cost connected with their practical evaluation are very high. Inorder to determine the optimal stacking operator for a certain sample in the zero-offset section, a simulta-neous search for at least two of the three CRS attributes is required. Therefore, we use equation (2) only forthe local optimization of the initial attributes, determined by means of a traveltime approximation which isless accurate but better suited for a global search. Such a traveltime approximation was presented by Chiraet al. (2001). We will rederive this formula in the following starting from equation (1). For this purpose,we establish in every pointX0 a local Cartesian coordinate system as depicted in Figure 15, with its originin X0 and itsx-axis being tangent to the surface inX0. In the local coordinate system thez-componentof any source or receiver point in the vicinity ofX0 can be approximated up to the second order by therelation

z(x) = −K0

2x2 , with K0 = −d

2z

dx2

∣∣∣∣∣∣∣∣∣X0

(3)

being the local curvature of the measurement surface inX0. Thus, in the local coordinate system thelocation of two pointsS andG in the vicinity ofX0 reads(

x(S)z(S)

)=(

mx − hx−K0

2 (mx − hx)2

)and

(x(G)z(G)

)=(

mx + hx−K0

2 (mx + hx)2

). (4)


Figure 15: Local Cartesian coordinate system with origin inX0 and x-axis tangent to the measurementsurface inX0.

For the corresponding midpoint and half-offset vectors we find, according to Figure 15,

~m(S,G) =(mx

mz

)= 1

2

(x(G) + x(S)z(G) + z(S)

)=(

mx

−K02 (m2

x + h2x)

), (5)

~h(S,G) =(hxhz

)= 1

2

(x(G)− x(S)z(G)− z(S)

)=(

hx−K0mxhx

). (6)

Denoting the dip angle of the measurement surface inX0 with respect to a globalz-axis byα0 we obtainfor ∆mz andhz after a simple rotation of the coordinate system

∆mx = mx cosα0 −mz sinα0 , hx = hx cosα0 − hz sinα0 , (7)

∆mz = mx sinα0 +mz cosα0 , hz = hx sinα0 + hz cosα0 . (8)

From equations (5), (6), and (7) we can derive

mx =∆mx

cosα0− sinα0

cosα0

K0

2(m2

x + h2x) and hx =

hxcosα0

− sinα0

cosα0K0mxhx. (9)

Multiplying the above equations bymx andhx, respectively, we find the second order approximation

m2x =

∆m2x

cos2 α0, h2

x =h2x

cos2 α0, and mxhx =

∆mxhxcos2 α0

. (10)

Together with equations (5), (6), (8) and (9), this leads to the following expressions

∆mz =sinα0∆mx

cosα0− K0

2 cos3 α0(∆m2

x + h2x) , (11)

hz =sinα0hxcosα0

− K0

cos3 α0∆mxhx . (12)

Inserting equations (11) and (12) into equation (1) and retaining only terms up to the second order yieldsthe parabolic CRS traveltime operator for smoothly curved topography. By squaring both sides and keepingagain only terms up to second order, we obtain the hyperbolic CRS traveltime operator for smooth curved


topography:

t2hyp(∆mx, hx) =(t0 +

2 ∆mx

v0 cosα0sin(β0 + α0)

)2

+2 t0 ∆m2

x

v0 cos2 α0

(KN cos2(β0 + α0)−K0 cos(β0 + α0)

)(13)

+2 t0 h2

x

v0 cos2 α0

(KNIP cos2(β0 + α0)−K0 cos(β0 + α0)

).

SettingK0 = 0 andα0 = 0 the above traveltime operator reduces to the well-known zero-offset CRSstack formula for a planar measurement surface (see, e. g. Höcht, 1998; Müller, 1999; Mann, 2002). Forconvenience this formula (in different notation) is repeated here. It reads

t2hyp,planar(∆mx, hx) =(t0 +

2v0

∆mx sinβ0

)2

+2 t0v0

KN cos2 β0 ∆m2x (14)

+2 t0v0

KNIP cos2 β0 h2x .

APPENDIX B

The redatuming of the zero-offset samples of stack and attribute sections is composed of a lateral displace-ment∆x and a time shift∆t. Denoting the emergence point of the zero-offset ray at the planar referencelevel byX ′

0(x′0, z

′0) the respective transformations read

x′0 = x0 + ∆x = x0 + (z′0 − z0) tanβ0 and t′0 = t0 + ∆t = t0 +2 (z′0 − z0)v0 cosβ0

. (15)

An important aspect to be considered by the redatuming procedure is that the stack amplitudes and es-pecially the CRS attributesKNIP andKN alter their values while the zero-offset ray is continued. Thepropagation law (see, e. g., Hubral and Krey, 1980) yields for the two wavefront radiiK ′

NIP andK ′N

measured at the planar reference level

K ′NIP =

(1

KNIP+

12

∆t v0

)−1

and K ′N =

(1KN

+12

∆t v0

)−1

(16)

The values of the emergence angle are not altered by the redatuming as no refraction occurs at the smoothlycurved reference level due to the conditionvr = v0. To map the stack amplitudes, thegeometrical spread-ing factorcalculated from the CRS attributes could be used to extrapolate appropriate values correspondingto the planar reference level. However, this is not yet implemented and the amplitudes are currently mappedwithout altering their values.

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